Existence and nonexistence of nontrivial solutions for a class of p-Kirchhoff type problems with critical Sobolev exponent

Abstract

In this work, by using variational methods we study the existence of nontrivial positive solutions for a class of p-Kirchhoff type problems with critical Sobolev exponent.

The Existence Result for a p -Kirchhoff-Type Problem Involving Critical Sobolev Exponent

“…It is worth mentioning that the first work on the Kirchhoff-type problem with critical Sobolev exponent is Alves et al in [3]. After that, many researchers dedicated to the study of several kinds of elliptic Kirchhoff equations with critical exponent of Sobolev in bounded domain or in the whole space ℝ N ; some interesting studies can be found in [4][5][6][7][8][9] and the references therein. More precisely, Naimen in [8] generalized the results of [10] to the semilinear Kirchhoff problem:…”

“…For larger dimensional case, Figueiredo in [5] considers the case N ≥ 3 if λ > 0 is sufficiently large. Matallah et al in [7] studied the existence and nonexistence of solutions for the following p-Kirchhoff problem:…”

“…Remark 2. If N ≥ p 2 , then max rp, N p − 1 / N − p = rp In the case where q ≤ rp, it is difficult to show that a Palais-Smale sequence of the corresponding energy is bounded; in this case, the authors in [7,9] used the truncation method to show the existence of solution under the condition " sufficiently large." Our objective in this paper is the existence of solution for all λ > 0.…”

Existence and multiplicity of solutions for (p,q)$$ \left(p,q\right) $$‐Laplacian Kirchhoff‐type fractional differential equations with impulses

“…In addition, Kirchhoff-type equations, which Kirchhoff [14] introduced in 1883 as a generalization of the well-known D'Alembert wave equation, are essential for the modeling of a variety of physical and biological systems. Some worthwhile studies of Kirchhoff equations include those by Graef et al [15], Hssini [16], Matallah et al [17], and references therein. It should be mentioned that the existence of solutions to the Kirchhoff-type fractional differential equations addressed by variational methods has also received much attention from scholars [18][19][20][21][22][23], as models based on fractional order are better suited to describing the memory and hereditary properties of many processes and materials.…”